Problem 38 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number divided by seven, plus two, is seventeen.

Step-by-Step Solution

Answer

The equation is \( \frac{x}{7} + 2 = 17 \).

Step 1: Identify the Unknown Number

The phrase "A number divided by seven" indicates that we need to identify this unknown number. Let's call this number \( x \).

Step 2: Translate the Division Phrase

The part "divided by seven" suggests that we should perform division on our unknown number \( x \). Thus, this becomes \( \frac{x}{7} \).

Step 3: Add Plus Two

Next, we have "plus two," meaning we add two to the result of the previous step. The expression now becomes \( \frac{x}{7} + 2 \).

Step 4: Set the Expression Equal to Seventeen

The phrase "is seventeen" indicates that the entire expression equals seventeen. Thus, we form the equation \( \frac{x}{7} + 2 = 17 \).

Step 5: Final Equation

The complete mathematical expression is \( \frac{x}{7} + 2 = 17 \).

Key Concepts

Translate Sentences to Equations

When faced with sentences that describe mathematical relationships, your first step is to transform the words into a math equation. This is like converting one language to another. Let's break it down:

• Identify keywords: Words like "is," "plus," "divided by," etc., are clues that signal mathematical operations.
• Determine the unknown: Decide what the unknown variable is, often represented by a symbol like \( x \).
• Create expressions: Combine the known numbers and operations with the unknown to build expressions.

For example, the sentence "A number divided by seven, plus two, is seventeen," involves finding a number (let's call it \( x \)) that fits this description. "Divided by seven" implies a division operation, "plus two" is addition, and "is seventeen" means the result equals 17. Consequently, the sentence translates to the equation \( \frac{x}{7} + 2 = 17 \).
Breaking sentences into parts using this method helps you clearly see the mathematical operations involved, allowing you to develop a corresponding equation.

Division in Equations

Understanding division is crucial, as it describes how a number is split into equal parts. In equations, division is often represented by the \( \div \) sign or as a fraction. When a problem involves division, you need to understand its role in the equation:

• Division distributes a number equally into parts; for example, "a number divided by 7" means splitting the number into 7 equal parts, represented mathematically as \( \frac{x}{7} \).
• It can often be a part of more complex expressions, where additional operations are performed on the quotient, such as adding or subtracting numbers.

In our example, "a number divided by seven" becomes the expression \( \frac{x}{7} \). This part of the sentence tells us how to mathematically represent the division component in equations.

Solving for Unknown Variables

Once you have translated your sentence into an equation, the next step is solving for the unknown variable located in the equation. Let's discuss how you can solve these equations step by step:

• Isolate the variable: Use inverse operations to get the variable by itself on one side of the equation.
• Perform arithmetic operations: In the equation \( \frac{x}{7} + 2 = 17 \), you need to subtract 2 from both sides to focus on the division part. You get \( \frac{x}{7} = 15 \).
• Eliminate the divisor: To solve \( \frac{x}{7} = 15 \), multiply both sides by 7 to undo the division, resulting in \( x = 105 \).

By isolating \( x \), you find that the unknown variable equals 105. Understanding how to reverse operations and maintain balance is key while solving. Each step involves simple arithmetic, but it requires careful handling to solve equations correctly.